Problem: Factor the following expression: $2$ $x^2$ $-3$ $x$ $-27$
Explanation: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(2)}{(-27)} &=& -54 \\ {a} + {b} &=& & & {-3} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-54$ and add them together. Remember, since $-54$ is negative, one of the factors must be negative. The factors that add up to ${-3}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-9}$ and ${b}$ is ${6}$ $ \begin{eqnarray} {ab} &=& ({-9})({6}) &=& -54 \\ {a} + {b} &=& {-9} + {6} &=& -3 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {2}x^2 {-9}x +{6}x {-27} $ Group the terms so that there is a common factor in each group: $ ({2}x^2 {-9}x) + ({6}x {-27}) $ Factor out the common factors: $ x(2x - 9) + 3(2x - 9) $ Notice how $(2x - 9)$ has become a common factor. Factor this out to find the answer. $(2x - 9)(x + 3)$